Misleading: JJJ BB's wearing BJJ BB's when they're not

Just something I've noticed in my area.

I had a BJJ instructor who was a purple belt at the time (now brown) and 3rd dan BB in traditional Jiu jitsu. When he was teaching the BJJ classes he would wear his black belt?! I find this very misleading. I've noticed this on other club websites (blue belt in BJJ and BB in JJJ but when teaching BJJ they wear there black blets).